To convert the quadratic equation $25x^2 - 20x - 4$ into its vertex form, we complete the square. Here's how it's done:

1. Factor out the coefficient of $x^2$ from the first two terms

Factor $25$ from $25x^2$ and $-20x$: $25x^2 - 20x - 4 = 25(x^2 - \frac{4}{5}x) - 4$

2. Complete the square

To complete the square inside the parentheses, find the term to add and subtract: $\left(\frac{\text{coefficient of } x}{2}\right)^2 = \left(\frac{-\frac{4}{5}}{2}\right)^2 = \left(\frac{-2}{5}\right)^2 = \frac{4}{25}$

Add and subtract $\frac{4}{25}$ inside the parentheses: $25\left(x^2 - \frac{4}{5}x + \frac{4}{25} - \frac{4}{25}\right) - 4$

Simplify: $25\left((x - \frac{2}{5})^2 - \frac{4}{25}\right) - 4$

3. Distribute the factor of $25$

$25(x - \frac{2}{5})^2 - 25 \cdot \frac{4}{25} - 4 = 25(x - \frac{2}{5})^2 - 4 - 4$

$25(x - \frac{2}{5})^2 - 8$

Final Vertex Form

The vertex form is: $25(x - \frac{2}{5})^2 - 8$

Vertex

The vertex of the parabola is $\left(\frac{2}{5}, -8\right)$.

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Do you have any questions, or would you like additional explanation?

5 Related Questions:

1. How do you identify the vertex directly from the standard form of a quadratic equation?
2. What is the geometric meaning of completing the square?
3. Can you derive the axis of symmetry from the vertex form?
4. How does the coefficient of $x^2$ affect the parabola's shape?
5. How do you find the x-intercepts from the vertex form?

Tip:

Completing the square is not only useful for rewriting quadratics but also for solving them when other methods are challenging!